Tottenham Hotspur are in the last eight of the Europa League but they arrived there with all the conviction of a drunk stumbling over the threshold as the bounteous inheritance of a 3-0 first-leg lead was all but squandered at the San Siro.

With Inter finishing 4-1 winners after extra time, Emmanuel Adebayor's close-range goal after six minutes of that overtime was enough to send Spurs through on away goals on a night given an unpleasant tinge by an inflatable banana being waved in the stand housing Inter's ultras.

Make no mistake, though, Spurs were misshapen from the start and cowed by Inter's ruthlessness in exposing a low-pressure midfield and high defensive line. A wonderfully dramatic game of football it may have been but the recording will make for chilling viewing for Tottenham's technophile manager.

"We took a risk with the shape," André Villas-Boas said. "We set out to score the goal that would have put us in a very comfortable position. We were the same as Inter were in London. We are not as bad as the performance showed. There are lots of positives and the desire was still there from the players to reach the last eight even in extra time."

Oh, Spurs. Villas-Boas deserves some slack in what is a very promising first season. But there are times when having a self-schooled, 35-year-old innovator in charge can lend an air of youthful impetus: and there are others where a charge of callowness is almost unavoidable. Villas-Boas will be taken to task by some for his tactics here as Tottenham played the opening hour of this match with a midfield so wide open you could hear the wind whistling through it beneath the open funnel of the Stadium Giuseppe Meazza's high concrete stands.

Driven on by a performance of fantastic movement and vision from Antonio Cassano, the best player on the pitch, Inter would have won this tie in normal time had Esteban Cambiasso's drive shaved the other side of Brad Friedel's post. It must be said a 4-0 victory would not have flattered Inter, for whom Cassano and the elegant Colombian playmaker Fredy Guarín seemed to have all the time in the world to look up and pick out the tempting green space behind Jan Vertonghen and William Gallas.

In Spurs' defence Inter were unrecognisable from the dispirited force of the first leg. And Villas-Boas had the presence to rouse his team in the break before extra time, drawing from a wilting group of players the burst of attacking vim that eventually decided the tie. And so Spurs are in the quarter-finals of a tournament that – pace the European Champions – contains no genuinely intimidating opponents and with the prospect of Gareth Bale, a palpable absence, returning.

If a first European trophy since the Tony Parks-inspired boys of 1984 looked some way off at the San Siro, this is still a club whose modern history has always had a tinge of glorious European nights about it, from the early days of Bill Nicholson's sorties into Europe that culminated in Cup-Winners' Cup triumph in 1963.

Here Spurs kicked off to a relatively mild welcome inside a quarter-full San Siro. It was deceptive. With Guarín conducting affairs in midfield Inter took a grip and on 19 minutes they took the lead. A simple pass inside Vertonghen by the right-back, Jonathan, found Palacio in space. His cross picked out Cassano unmarked at the back post and he headed the ball down on to the turf and up into the top corner.

Five minutes later Spurs were again undone by a single pass, this time a lovely dink from Cassano for the galloping Palacio, whose attempted lob hit the top of the bar.

Inter's second came six minutes after half-time as Cambiasso played a sweet pass inside Kyle Walker to Palacio. His finish, low past Friedel, was never in doubt. Suddenly Inter were only a goal behind with all the momentum in the tie and a newly awakened curva nord providing boisterous support.

Villas-Boas responded by replacing Jermain Defoe with Lewis Holtby to stiffen that porous midfield but the equaliser in the tie arrived on 75 minutes. Cassano's low, whipped free-kick sneaked through a copse of legs but was going wide before it was deflected in off Gallas's heel.

And so to extra time and the moment that Spurs finally cashed in that first-leg lead. Dembélé's low shot was palmed out by Samir Handanovic and Adebayor's long right leg stretched to force the ball home. There was still a twist as Ricardo Alvarez headed home to make it 4-1 on the night.